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Public-Key Cryptography Public-key Cryptogra; Theory and Practice ABHIJIT DAS Department of Computer Science and Engineering Indian Institute of Technology Kharagpur C. E. VENIMADHAVAN Department of Computer Science and Automation Indian Institute ofScience, Bangalore PEARSON Chennai • Delhi • Chandigarh Upper Saddle River • Boston • London Sydney • Singapore • Hong Kong • Toronto • Tokyo Contents Preface xiii Notations xv 1 Overview 1 1.1 Introduction 2 1.2 Common Cryptographic Primitives 2 1.2.1 The Classical Problem: Secure Transmission of Messages 2 Symmetric-key or secret-key cryptography 4 Asymmetric-key or public-key cryptography 4 1.2.2 Key Exchange 5 1.2.3 Digital Signatures 5 1.2.4 Entity Authentication 6 1.2.5 Secret Sharing 8 1.2.6 Hashing 8 1.2.7 Certification 9 1.3 Public-key Cryptography 9 1.3.1 The Mathematical Problems 9 1.3.2 Realization of Key Pairs 10 1.3.3 Public-key Cryptanalysis 11 1.4 Some Cryptographic Terms 11 1.4.1 Models of Attacks 12 1.4.2 Models of Passive Attacks 12 1.4.3 Public Versus Private Algorithms 13 2 Mathematical Concepts 15 2.1 Introduction 16 2.2 Sets, Relations and Functions 16 2.2.1 Set Operations 17 2.2.2 Relations 17 2.2.3 Functions 18 2.2.4 The Axioms of Mathematics 19 Exercise Set 2.2 20 2.3 Groups 21 2.3.1 Definition and Basic Properties . 21 2.3.2 Subgroups, Cosets and Quotient Groups 23 2.3.3 Homomorphisms 25 2.3.4 Generators and Orders 26 2.3.5 Sylow's Theorem 27 Exercise Set 2.3 29 2.4 Rings 31 2.4.1 Definition and Basic Properties 31 2.4.2 Subrings, Ideals and Quotient Rings 34 2.4.3 Homomorphisms 37 2.4.4 Factorization in Rings 39 Exercise Set 2.4 42 2.5 Integers 44 iv Contents 2.5.1 Divisibility 44 2.5.2 Congruences 45 2.5.3 Quadratic Residues 48 2.5.4 Some Assorted Topics 52 The prime number theorem 53 Density of smooth integers 54 The extended Riemann hypothesis 54 Exercise Set 2.5 56 2.6 Polynomials 57 2.6.1 Elementary Properties 58 2.6.2 Roots of Polynomials 59 2.6.3 Algebraic Elements and Extensions 61 Exercise Set 2.6 63 2.7 Vector Spaces and Modules 64 2.7.1 Vector Spaces 65 2.7.2 Modules 69 2.7.3 Algebras 71 Exercise Set 2.7 72 2.8 Fields 74 2.8.1 Properties of Field Extensions 74 2.8.2 Splitting Fields and Algebraic Closure 76 2.8.3 Elements of Galois Theory 78 Exercise Set 2.8 79 2.9 Finite Fields 80 2.9.1 Existence and Uniqueness of Finite Fields 80 2.9.2 Polynomials over Finite Fields 82 2.9.3 Representation of Finite Fields 85 Exercise Set 2.9 88 2.10 Affine and Projective Curves 90 2.10.1 Plane Curves 90 2.10.2 Polynomial and Rational Functions on Plane Curves 92 2.10.3 Maps Between Plane Curves 95 2.10.4 Divisors on Plane Curves 95 Exercise Set 2.10 97 2.11 Elliptic Curves 98 2.11.1 The Weierstrass Equation 98 2.11.2 The Elliptic Curve Group 101 2.11.3 Elliptic Curves over Finite Fields 106 Exercise Set 2.11 107 2.12 HyperellipticCurves Ill 2.12.1 The Defining Equations Ill 2.12.2 Polynomial and Rational Functions 112 2.12.3 TheJacobian 115 Exercise Set 2.12 118 2.13 Number Fields 119 2.13.1 Some Commutative Algebra 120 Ideal arithmetic 120 Localization 120 Integral dependence 121 Contents v Noetherian rings 1^J Dedekind domains 125 2.13.2 Number Fields and Rings 125 2.13.3 Unique Factorization of Ideals 131 2.13.4 Norms of Ideals 135 2.13.5 Rational Primes in Number Rings 137 139 2.13.6 Units in a Number Ring Exercise Set 2.13 139 2.14 p-adic Numbers 143 2.14.1 The Arithmetic of p-adic Numbers 143 2.14.2 The p-adic Valuation 145 2.14.3 Hensel's Lemma 149 Exercise Set 2.14 151 2.15 Statistical Methods I54 2.15.1 Random Variables and Their Probability Distributions 154 2.15.2 Operations on Random Variables 155 2.15.3 Expectation, Variance and Correlation 159 2.15.4 Some Famous Probability Distributions 162 ' Uniform distribution 162 Bernoulli distribution 163 Normal distribution 163 2.15.5 Sample Mean, Variation and Correlation 164 Exercise Set 2.15 165 3 Algebraic and Number-theoretic Computations 173 3.1 Introduction 174 3.2 Complexity Issues 174 3.2.1 OrderNotations 175 3.2.2 Randomized Algorithms 177 3.2.3 Reduction Between ComputationalProblems 178 Exercise Set 3.2 179 3.3 Multiple-precision Integer Arithmetic 180 3.3.1 Representation of Large Integers 181 3.3.2 Basic Arithmetic Operations 181 Addition and subtraction 182 Multiplication 183 Squaring 184 Fast multiplication 184 Division 185 Bit-wise operations 187 3.3.3 GCD . 187 3.3.4 Modular Arithmetic 190 Modular exponentiation 190 Montgomery exponentiation 192 Exercise Set 3.3 193 3.4 Elementary Number-theoretic Computations 195 3.4.1 Primality Testing '. 195 Deterministic . primality proving , 197 3.4.2 Generating Random Primes 199 vi Contents 3.4.3 Modular Square Roots 200 Exercise Set 3.4 201 3.5 Arithmetic in Finite Fields 204 3.5.1 Arithmetic in the Ring F2 [X] 204 3.5.2 Finite Fields of Characteristic 2 208 3.5.3 Selecting Suitable Finite Fields 210 3.5.4 Factoring Polynomials over Finite Fields 212 Square-free factorization 212 Distinct-degree factorization 213 Equal-degree factorization 214 Exercise Set 3.5 215 3.6 Arithmetic on Elliptic Curves 218 3.6.1 Point Arithmetic 218 3.6.2 Counting Points on Elliptic Curves 219 The SEA algorithm 219 The Satoh-FGH algorithm 221 3.6.3 Choosing Good Elliptic Curves 223 3.7 Arithmetic on Hyperelliptic Curves 224 3.7.1 Arithmetic in the Jacobian 225 3.7.2 Counting Points in Jacobians of Hyperelliptic Curves 225 Exercise Set 3.7 228 3.8 Random Numbers 228 3.8.1 Pseudorandom Bit Generators 228 3.8.2 Cryptographically Strong Pseudorandom Bit Generators 229 3.8.3 Seeding Pseudorandom Bit Generators 230 Exercise Set 3.8 231 4 The Intractable Mathematical Problems 237 4.1 Introduction 238 4.2 The Problems at a Glance 239 Exercise Set 4.2 242 4.3 The Integer Factorization Problem 243 4.3.1 Older Algorithms 244 Trial division 244 Pollard's rho method 244 Pollard's p-1 method 245 Williams' p + 1 method 247 4.3.2 The Quadratic Sieve Method 248 The basic algorithm 248 Sieving 249 Incomplete sieving 251 Large prime variation 251 The multiple polynomial quadratic sieve 252 Parallelization 253 TWINKLE: Shamir's factoring device 254 4.3.3 Factorization Using Elliptic Curves 255 4.3.4 The Number Field Sieve Method 258 Selecting the polynomial f(X) 259 Construction of Q 259 Contents vii Construction of Q 259 Construction of U 260 260 Computing the factorization of a + ba Sieving 261 The running time of the SNFSM 262 Exercise Set 4.3 262 4.4 The Finite Field Discrete Logarithm Problem 264 4.4.1 Square Root Methods 264 Shanks'baby-step-giant-step method 265 Pollard's rho method 265 The Pohlig-Hellman method 266 4.4.2 The Index Calculus Method 267 4.4.3 Algorithms for Prime Fields 268 The basic ICM 268 The linear sieve method 270 The number field sieve method 272 4.4.4 Algorithms for Fields of Characteristic 2 273 The basic ICM 274. The adaptation of the linear sieve method 275 Coppersmith's algorithm 276 Exercise Set 4.4 279 4.5 The Elliptic Curve Discrete Logarithm Problem (ECDLP) 281 4.5.1 The MOV Reduction 282 The correctness of the algorithm 283 Choosing fc 284 Computing em{P,R) 284 4.5.2 The SmartASS Method 286 4.5.3 The Xedni Calculus Method 289 Exercise Set 4.5 291 4.6 The Hyperelliptic Curve Discrete Logarithm Problem 292 4.6.1 Choosing the Factor Base 293 4.6.2 Checking the Smoothness of a Divisor 293 4.6.3 The Algorithm 294 4.7 Solving Large Sparse Linear Systems over Finite Rings 294 4.7.1 Structured Gaussian Elimination 296 4.7.2 The Conjugate Gradient Method 297 4.7.3 The Lanczos Method 298 4.7.4 The Wiedemann Method 299 4.8 The Subset Sum Problem 300 4.8.1 The Low-Density Subset Sum Problem 301 4.8.2 The Lattice-Basis Reduction Algorithm 302 Exercise Set 4.8 304 5 Cryptographic Algorithms 309 5.1 Introduction 310 5.2 Secure Transmission of Messages 310 5.2.1 The RSA Public-key Encryption Algorithm 310 RSA key pair 310 RSA encryption 312 viii Contents RSA decryption 312 5.2.2 The Rabin Public-key Encryption Algorithm 313 Rabin key pair 313 Rabin encryption 314 Rabin decryption 314 5.2.3 The Goldwasser-Micali Encryption Algorithm 315 Goldwasser-Micali key pair 315 Goldwasser-Micali encryption 315 Goldwasser-Micali decryption 316 5.2.4 The Blum-Goldwasser Encryption Algorithm 317 Blum-Goldwasser key pair 317 Blum-Goldwasser encryption 318 Blum-Goldwasser decryption 318 5.2.5 The ElGamal Public-key Encryption Algorithm 319 ElGamal key pair 319 ElGamal encryption 320 ElGamal decryption 320 5.2.6 The Chor-Rivest Public-key Encryption Algorithm 321 Chor-Rivest key pair 321 Chor-Rivest encryption 322 Chor-Rivest decryption 322 5.2.7 The XTR Public-key Encryption Algorithm 323 XTR key pair 327 XTR encryption 327 XTR decryption 328 5.2.8 The NTRU Public-key Encryption Algorithm 328 NTRU key pair 328 NTRU encryption 330 NTRU decryption 331 Exercise Set 5.2 332 5.3 Key Exchange 334 5.3.1 Basic Key-Exchange Protocols 334 The Diffie-Hellman key-exchange protocol 334 Small-subgroup attacks 335 Cofactor exponentiation 336 5.3.2 Authenticated Key-Exchange Protocols 336 Unknown key-share attacks 336 The Menezes-Qu-Vanstone key-exchangeprotocol 338 Exercise Set 5.3 339 5.4 Digital Signatures 340 5.4.1 The RSA Digital Signature Algorithm 341 5.4.2 The Rabin Digital Signature Algorithm 342 5.4.3 The ElGamal Digital Signature Algorithm 343 5.4.4 The Schnorr Digital Signature Algorithm 344 5.4.5 The Nyberg-Rueppel Digital Signature Algorithm 345 5.4.6 The Digital Signature Algorithm (DSA) 346 5.4.7 The Elliptic Curve Digital Signature Algorithm (ECDSA) 348 5.4.8 The XTR Signature Algorithm 349 5.4.9 The NTRUSign Algorithm 352 Contents ix 5.4.10 Blind Signature Schemes 355 Chaum's RSA blind signature
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